Question

geometry a - g semester b 25 - 26
angle relationships
what is the measure of circumscribed \\( \angle x \\)?
\\( 45 ^ { \circ } \\)
\\( 95 ^ { \circ } \\)
\\( 50 ^ { \circ } \\)
\\( 90 ^ { \circ } \\)

Explanation:

Step1: Recall the property of the angle between a tangent and a radius

A tangent to a circle is perpendicular to the radius at the point of tangency. So, $\angle VYX = 90^{\circ}$ and $\angle VWX=90^{\circ}$.

Step2: Use the sum of angles in a quadrilateral

The sum of the interior angles of a quadrilateral is $360^{\circ}$. In quadrilateral $VYXW$, we know $\angle VYX = 90^{\circ}$, $\angle VWX = 90^{\circ}$, $\angle YVW=90^{\circ}$ (given). Let $\angle X$ be the unknown angle.
Using the formula $\angle VYX+\angle VWX+\angle YVW+\angle X = 360^{\circ}$.
Substitute the known values: $90^{\circ}+90^{\circ}+90^{\circ}+\angle X=360^{\circ}$.
Simplify the left - hand side: $270^{\circ}+\angle X = 360^{\circ}$.
Solve for $\angle X$: $\angle X=360^{\circ}- 270^{\circ}$.

Answer:

$90^{\circ}$